Using Figure 3, it is seen that the covariates that have the strongest relationship with SBP are smoke, exercise, weight, height, overweight, alcohol, TRT, and BMI. The other covariates do not seem to have that strong of a relationship with SBP. BMI is strongly correlated with height, weight and overweight which raises a multicollinearity problem if it were to be added to the model. Ideally would be better to remove it.Main Effect Full Model
Figure 4. VIF values of all covariates
## gender married smoke exercise age weight height overwt
## 5.443472 1.025684 1.109423 1.041756 1.026504 22.872725 14.952780 5.109263
## race alcohol trt bmi stress salt chldbear income
## 1.084261 1.079912 1.060084 34.883669 1.046019 1.071238 5.276208 1.068000
## educatn
## 1.044346Variance Inflation Factor
Looks like there is a multicollinearity problem with weight, height, and BMI. Gender and chldbear seems to have a VIF about 5~ and could pose as a problem. Removing BMI could be beneficial to get a more accurate model. In addition, BMI = \(weight/height^2\), and that a higher BMI depending on the gender, determines if a person is overweight or not. (CDC,2021) The data seems like it is using BMI to determine someone is overweight or not.
Doing the ANOVA test, shows that reduced model has the same effect as the full model as the p-value > 0.05. Thus we can continue to do some of the residual diagnostics with this model.
Figure 4. VIF values of all covariates witout BMI
## gender married smoke exercise age weight height overwt
## 5.326526 1.023309 1.107292 1.041521 1.026344 3.629298 2.616850 5.057393
## race alcohol trt stress salt chldbear income educatn
## 1.083517 1.074089 1.059565 1.045077 1.071100 5.166147 1.066463 1.043999As we can see, there are no VIF values greater than 10 anymore. There is no indication of serious multicollinearity anymore.
This model with coefficients in Figure 4. includes all predictors, excluding BMI. Since BMI could cause a multicollinearity issue, it was removed. The model here is significant with p-value < 0.05. Meaning that not all values of the coefficients are 0. The coefficients for trt, smoke, exercise and alcohol seems to be significant factors for the model with p-values < 0.05. The intercept standard is quite high as well, at 58.8365, in comparison to the intercept itself at 69.08773. Using the F-test to compare the model with BMI and without BMI, it results in a p-value of 0.7715. Which means that BMI does not make a significant impact on the model predicting SBP.
Residual diagnostics of the main effect model
The residuals values of this model look normally distributed in reference to Figure 5. and Figure 6.. The Shapiro-Wilk test gives a P-Value of 0.5881, therefore failing to reject the null hypothesis of normality. However, there is a concern with the variance of the residuals, which they are not constant. There seems to be some resemblance of a trumpet shape for figure 7. with the points, which puts the constant variance of the error terms into question. To diagnose this, the Breusch-Pagan test, gives a P-value of 0.01755 therefore rejecting the null hypothesis that the variance of the residual is constant. As a remedial measure, a box cox transformation is done below.
shapiro.test(bp_full2$residuals)
bptest(bp_full2, studentize = F) Transformed full model
Using a box-cox transformation, let \(\lambda\) = 0.5454545 onto SBP values.
Residual Diagnosis for the Transformed model
Visually, using Figure 9, and Figure 10, it shows that the residuals are likely from a normally distributed sample. Verifying speculations of the normality of the error terms, Shapiro-Wilk test gives a p-value of 0.4015 therefore failing to reject the null hypothesis where the residuals are not from a normally distributed sample with a significance level of 0.05. Providing strong evidence that the residuals are from a normally distributed sample. Although figure 11. shows that the variance of the error terms may not look constant with the resemblance of a trumpet shape with the plots the Breusch-Pagan test, gives us a p-value of 0.06193. Thus, failing to reject the null hypothesis that the variance of the error terms are constant with a significance level of 0.05 providing evidence that the residuals have constant variance.
Backwards elimination
AIC/BIC
Using backwards elimination as described in the lecture with significance level of =0.20, (through the function stepAIC() in R) for the procedure, gives us a new model dropping married, age, height, overwt, race, stress, salt ,income, educatn. The AIC for the new model is 1936.76, in comparison to 1950.57 for the full model. Using the F-test to compare both models, with a p-value of 0.9117, which results in a failure to reject the null hypothesis that the reduced model and full model have the same effect. In other words, the reduced model is as effective as the full model.
Residual Diagnostics for Reduced model
The residuals values of this model look normally distributed in reference to Figure 12. and Figure 13.. The Shapiro-Wilk test gives a P-Value of 0.5881, therefore failing to reject the null hypothesis of normality. However, there is a concern with the variance of the residuals, which are not constant. There seems to be some resemblance of a trumpet shape for figure 7. with the points, which puts the constant variance of the error terms into question. To diagnose this, the Breusch-Pagan test, gives a P-value of 0.01755 therefore rejecting the null hypothesis that the variance of the residual is constant. As a remedial measure, a box cox transformation is done below.
Residual diganostics for Transformed reduced model
Figure 15.
Transformed reduced model
Again, using a box-cox transformation, = 1.030303. Figure 15. show the box cox plot for the reduced model, in which the maximum lambda is 1.030303. Using the equation from Figure 8b. to apply the transformation on the SBP values in the data set. Then use this new transformed data to regress over the same covariates.
Residual diagnosis for transformed reduced model
Similarly to the past residual diagnosis, the graphs give the same result, figure 16 and figure 17, showing that the residuals are from a normally distributed sample. However, figure 18. shows that there could be some issue with error variance being constant. Using Shapiro-Wilk test to verify the normality, since P-value as 0.4873 is greater than 0.05, therefore failing to reject the null hypothesis that the residuals are from a normal sample. With the Breusch-Pagan test, the p-value is 0.002354, which is less than 0.05, thus rejecting the null hypothesis that the error terms have constant variance.
Remedial measures: Weighted least squares
In figure 19. residual values of the fitted weight least squares model are plotted against weight. There seems to be a random scatter of points on this plot, verifying that the residuals do not depends on this covariate. In Figure 20. the whiskers of the box-plot for treatment, seems to give the residuals more variance one with, than one without. Chldbear, follows a similar pattern where the whiskers are slowly shortening but slightly. In addition, the rest of the box-plots shows more of a constant variance of the residuals.
Graphical Diagnostics for influence
## NULLols_plot_dfbetas(mod9, print_plot =T)
The DFBETAS plots show that observations 53 seems to give the influence above the threshold on all the covariates. In addition, trt has the least amount of observations above the threshold for DFBETAS and exercise has the most above the threshold. Observations 53 for exercise seems to affect the covariate trt a lot more than the other observations. Alcohol seems to be the least affected by observations 53.
ggarrange(p2, p3,p4,p5)
Observing Cooks D chart and the influence diagnostic plots, it can be seen that the most influential points are 25,35, 53. However, overall there are many influential observations such that if these influential points were to be gotten rid of, there is a possiblity would be another set of them after with the way they are scattered.
The outlier and leverage diagnostics plot. shows that observation 53 has leverage and is an outlier. Outlier values in this figure shows that it is in a triangular shape. When ridding the data of these influential points there is a possibility of another set of outliers being present. The deleted studentized residual vs. Predictor plot also has an abundance of outliers. Again, removing these outliers could create another set of outliers here. Therefore, removing these outliers would probably not have much effect on the data set.
Cross validation for main effect model
## WLS Model Reduced Model Full Model
## MSE 1.6670 628.9000 630.8000
## MSPR 745.4301 720.4247 674.2833The WLS model seems very likely to be over fitting the data, given a very low MSE of 1.667, in comparison to the MSPR of 745.4301. In addition, the reduced model has a MSPR of 720.4247 and a MSE of 628.9. While the full model has an MSE of 630.8 and a MSPR of 674.2833. The reduced model would be considered less valid than the full model since there is a greater difference between MSPR and MSE with the reduced model than the full model.